Chapter 2: `Diffraction <CH2_00-Diffraction.ipynb>`__

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Fourier Transform Laboratory

Real and Reciprocal Lattice

Download

part of

`MSE672: Introduction to Transmission Electron Microscopy <../_MSE672_Intro_TEM.ipynb>`__

by Gerd Duscher, Spring 2021

Microscopy Facilities Joint Institute of Advanced Materials Materials Science & Engineering The University of Tennessee, Knoxville

Background and methods to analysis and quantification of data acquired with transmission electron microscopes.

Import numerical and plotting python packages

Check Installed Packages

import sys
from pkg_resources import get_distribution, DistributionNotFound

def test_package(package_name):
    """Test if package exists and returns version or -1"""
    try:
        version = get_distribution(package_name).version
    except (DistributionNotFound, ImportError) as err:
        version = '-1'
    return version

# Colab setup ------------------
if 'google.colab' in sys.modules:
    !pip install pyTEMlib -q
# pyTEMlib setup ------------------
else:
    if test_package('pyTEMlib') < '0.2021.1.9':
        print('installing pyTEMlib')
        !{sys.executable} -m pip install  --upgrade pyTEMlib -q
# ------------------------------
print('done')

done

Load the plotting and figure packages

import sys
if 'google.colab' in sys.modules:
    %pylab --no-import-all inline
else:
    %pylab --no-import-all notebook

Populating the interactive namespace from numpy and matplotlib

Fourier Transform Input

length = 256
half = int(length/2)
x_axis  = np.arange(length)-half
real_space = np.zeros(length)
real_space[half] = 0.
real_space[half-10] = 0.
real_space[half+10-1] = 0.
real_space[half-120] = 1.
real_space[half+120] = 0.


for i in range(int(length/4)):
#    if i % 2:
#        real_space[i*4] = 1.
#    else:
#        real_space[i*4] = .5
    pass

plt.figure()
plt.plot(x_axis, real_space);

Fourier Transform Output

real_space = np.zeros(length)
real_space[half] = 0.
real_space[half-10] = 0.
real_space[half+10-1] = 0.
real_space[half-127] = 0.
real_space[half+127] = 1.

fourier_space = np.fft.fft(real_space)

fig, (ax1, ax2) = plt.subplots(1, 2, sharey=True)

ax1.plot(x_axis, real_space.real , label='real')
ax1.plot(x_axis, real_space.imag , label='imag')
ax1.legend()
ax1.title.set_text('Real Space')

ax2.plot(x_axis, fourier_space.real , label='real')
ax2.plot(x_axis, fourier_space.imag , label='imag')
ax2.legend()
ax2.title.set_text('Reciprocal Space');

Inverse Fourier Transform

fourier_space.imag = 0
real_space_2 = np.fft.ifft(fourier_space)

fig, (ax1, ax2) = plt.subplots(1, 2, sharey=True, dpi=160)
ax1.plot(x_axis, fourier_space.real , label='real')
ax1.plot(x_axis, fourier_space.imag , label='imag')
ax1.title.set_text('Reciprocal Space')
ax2.plot(x_axis, real_space_2.real , label='real')
ax2.plot(x_axis, real_space_2.imag , label='imag')
ax2.title.set_text('Real Space')
ax2.legend();

And now in 3D

The Fourier Transform of the sample shape is a major contributer to the possible excitation error

from skimage import measure
from mpl_toolkits.mplot3d import Axes3D

##################
# Make a disk
##################
s = 32
vol = np.zeros((32*2,32*2,32*2))

# Make one circle
circle = np.zeros((64,64))
for x in range (-32,32):
    for y in range(-32,32):
        if x**2+y**2 < s**2-100:
            circle[x+32,y+32]=1
# Make a disk by copying the full circle
thickness = 10
for z in range(int(s-thickness/2),int(s+thickness/2)):
    vol[:,:,z] = circle

# Make a cube
s = 32
#vol = np.zeros((32*2,32*2,32*2))
width = 20
vol[s-width:s+width,s-width:s+width,s-width:s+width] = 1

# Fourier Transform in 3D
ft = np.fft.fftshift(np.fft.fftn(vol))

#Make surfaces from volume
verts, faces, _, _ = measure.marching_cubes_lewiner(np.absolute(ft), 400.)
rverts, rfaces, _, _ = measure.marching_cubes_lewiner(vol, 0)


fig = plt.figure(figsize=(10, 5))
ax1 = fig.add_subplot(121, projection='3d')
ax1.plot_trisurf(rverts[:, 0], rverts[:,1], rfaces, rverts[:, 2], lw=1)
ax1.set_xlim(0, 64)
ax1.set_ylim(0, 64)
ax1.set_zlim(0, 64)
ax1.set_title('Real Space')

ax2 = fig.add_subplot(122, projection='3d')
ax2.plot_trisurf(verts[:, 0], verts[:,1], faces, verts[:, 2], lw=1)
ax2.set_xlim(0, 64)
ax2.set_ylim(0, 64)
ax2.set_zlim(0, 64)
ax2.set_title('Reciprocal Space');

Conclusions

Fourier transform of symmetric function do not have a imaginary component.

Imaginary Fourier components are often associated with absorption. Why?

Back: Atomic Form Factor

Next: Structure Factors

Chapter 2: Diffraction

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